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Britain could leave the EU if the results of two new polls are translated into votes in a referendum - reported the SkyNews.
One gives a narrow lead in favour of a British exit and the other is split almost 50-50 once undecided voters have been discounted.
They come amid indications that David Cameron will struggle to get an agreement on some of his four key demands for EU reform.
The Prime Minister has promised to let the British people vote on the reforms in an in-out referendum by the end of 2017.
Legislation paving the way for the historic referendum cleared Parliament on Monday night after peers backed down from an attempt to allow 16 and 17-year-olds the vote.
Number 10 has refused to rule out a "plan B" if Mr Cameron fails to win European support for his plan to stop EU migrants receiving benefits for four years after arriving in the UK.
When an ICM poll asked people, if they were voting tomorrow, should Britain stay in or leave the EU, pollsters found 42% of people would vote to remain while 41% would vote to leave, with 17% undecided.
The poll was commissioned by the Vote Leave campaign, which wants the UK to quit the EU.
The poll suggests the presence of measures to curb immigration from within the EU will be a key factor in whether Brexit goes ahead.
If there are no plans to control movement within the EU when the vote takes place, 45% say they would vote to leave the EU, 40% say they would vote to remain, with 15% undecided, the poll found.
A separate Survation poll found that 42% want to leave the EU and 40% want to remain, with 18% undecided.
Once the undecided voters had been excluded, 51% wanted to leave and 49% wanted to retain links with Brussels, according to the poll in the Daily Express.
It was commissioned by the Alliance for Direct Democracy in Europe, a European political party which includes UKIP.
Survation asked more than 10,000 adults in Britain on 3 December.
Vote Leave campaign director Dominic Cummings said: "The failures of the EU project over the past decade have sapped the enthusiasm of its supporters.
"Every day more people see that the EU is undermining prosperity, security, and democracy and they are increasingly concluding that it's safer to take back control than to keep handing over powers."